Is isotropic turbulence decay governed by asymptotic behavior of large scales? An eddy-damped quasi-normal Markovian-based data assimilation study

The present paper deals with the identification of the scales and features of the initial kinetic energy spectrum that govern the decay regime of freely decaying homogeneous isotropic turbulence (HIT). To this end, a Data Assimilation (DA) study is performed, which is based on a variational optimal control problem with the eddy-damped quasi-normal Markovian (EDQNM) model whose adjoint equation is derived in the present work. The DA procedure consists in reconstructing the initial kinetic energy spectrum in order to minimize the error committed on some features of decaying turbulence with respect to a targeted EDQNM simulation. The present results show that the decay of HIT over finite time is governed by a finite range of large scales, i.e., the scales ranging from the initial to the final integral scales (or equivalently by wave numbers comprised between the initial and the final location of the peak of the energy spectrum). The important feature of the initial condition is the slope of the energy spectrum at these scales, if such a slope can be defined. This is coherent with previous findings dealing with decay of non-self-similar solutions, or with the key assumptions that underly the Comte-Bellot–Corrsin theory. A consequence is that the finite time decay of HIT is not driven by the asymptotic large-scale behavior of the energy spectrum E(k → 0, t = 0) or the velocity correlation function f(r → +∞, t = 0), or even scales such as kL ≪ 1 or L/r ≪ 1. Governing scales are such that kL(t) = O(1) for values of the integral scale L(t) observed during the finite time decay under consideration. As a matter of fact, a null sensitivity of finite time decay of turbulence with respect to the asymptotic large scale features of the initial condition is observed. Therefore, the asymptotic features of the initial condition should not be investigated defining an inverse problem based of features of turbulence decay observed over a finite time.